Coriolis Meter

ABSTRACT

In accordance with example embodiments of the present disclosure, a method for determining parameters for, and application of, models that correct for the effects of fluid inhomogeniety and compressibility on the ability of Coriolis meters to accurately measure the mass flow and/or density of a process fluid on a continuous basis is disclosed. Example embodiments mitigate the effect of multiphase fluid conditions on a Coriolis meter.

TECHNICAL FIELD

The present disclosure relates to apparatus or methods for measuring themass flow and/or density of fluids wherein the fluid flows through themeter in continuous flow and to flow meters and flow measurement,Coriolis meters that employ multiple frequency oscillations and sonarflow measurement.

BACKGROUND

A Coriolis flow meter measures a parameter of a fluid, including but notlimited to parameters like the mass flow and/or density of a fluidthrough a conduit. The same Coriolis meter can measure fluids of varyingdensities, flow rates, temperatures, phases and viscosity. The mass flowrate is the mass of a fluid moving past a given point per unit time.Volumetric flow rate is the volume of fluid moving past a given pointper unit time. Coriolis meters report volumetric flow by dividing themeasured mass flow rate by the measured fluid density. Coriolis metersare often used to measure a wide variety of fluids, including liquid andgases and mixtures of the liquid and gases and fluid conveyed solids.Coriolis meters are known as highly accurate mass and density meters forhomogeneous fluids and fluids with low compressibility. However, asunderstood in the art, the accuracy of a Coriolis meter degrades withthe introduction of inhomogeneities in the process-fluid and increasesin process-fluid compressibility. For example, in the introduction ofentrained gases within a liquid, and other types of multiphase flows,introduces both inhomogeneities and increases the compressibility, andis known to cause errors in reported mass flow and/or process-fluiddensity

Coriolis meters measure the mass flow and density of the process-fluidby measuring the influence that the process-fluid has on the vibrationalcharacteristics of vibrating, fluid-conveying, flow tubes. The influenceof the process-fluid on the vibrational characteristics of the vibratingflow tubes depends on the homogeneity and the compressibility of theprocess-fluid. For a homogeneous, highly-incompressible fluid, thecenter of mass of the fluid essentially follows the center of mass ofthe vibrating, fluid-conveying flow tubes. In this case, the mass flowof the process fluid can be calibrated to be essentially proportional toa measured twist, or deformation, of the vibrational mode shape of theflow tube. The density of the process fluid can be calibrated to achange in the natural frequency of the fluid-conveying flow tube.

However, as fluid inhomogeneity and/or compressibility increase, thecenter-of-mass of the process-fluid increasingly departs from thecenter-of-mass of the flow tube when vibrated. This departure changesthe relationship between said measured vibrational characteristics andsaid fluid properties, thereby resulting in a Coriolis meter, calibratedon essentially homogeneous, incompressible fluids, to incorrectlyinterpret the mass flow and/or density of a process-fluid with increasedcompressibility and/or inhomogeneity. Decoupling of an inhomogeneousfluid is said to occur within a fluid when one phase of a fluid vibratesdifferently than another phase. Bubbly liquids are an example of a broadclass of fluids which can exhibit large and variable amounts of bothdecoupling due to inhomogeneities and compressibility.

Sonar flow meters can measure the speed at which sound propagatesthrough a fluid contained in a fluid-conveying conduit. Sonar flowmeasurement is effective in measuring the speed of sound in wide rangeof single and multiphase mixtures of gas and liquids, including fluidswith conveyed solids. Sonar flow meter are also effective in measuringthe speed of sound in bubbly mixtures and utilizing this speed of soundmeasurement in the determination of gas void fraction of bubbly fluids.

One skilled in the art understands that process fluid variability refersto the inhomogeneity of a fluid or the compressibility of a fluid, orany other fluid variability understood in the art.

Some improvements have been made in the application of Coriolis meterson multiphase flows by measuring the drive gain of tube vibration. Thedrive gain of a Coriolis meter is a measure of the oscillatory forcerequired to vibrate the fluid-conveying flow tube to a prescribedamplitude. Since the introduction of entrained gases increases thedamping of the vibrational mode of the fluid-conveying flow tubes, thedrive gain of a Coriolis meter often correlates with the amount ofentrained gas. Although this approach is often a reliable indicator ofthe presence of entrained gas, it does not typically reliably quantifythe amount of entrained gas, nor has it been successfully used tocorrect the errors in the reported mass flow or density measurements dueto the entrained gas.

In one example within the state of the art, a dual tube Coriolis meterhas flow tubes driven at two different vibrational modes, each with adifferent frequency. The meter provides a measure of the density of theprocess fluid by measuring the natural frequency of each of the twomodes of the fluid-conveying flow tubes and interpreting the naturalfrequency in terms of the density of the process-fluid utilizing acalibration determined for an essentially incompressible, homogeneousfluid. In this method, the difference in the measured densities isinterpreted as a measure of the influence of the entrained gas presentin the process-fluid. This difference in conjunction with the differencein vibrational frequency is used to estimate the density of the liquidwithout the entrained gas.

The state of the art has yet to effectively quantify the combinedeffects of decoupling (due to inhomogenieties) and compressibility onthe mass flow and/or density of a process-fluid reported by amultifrequency Coriolis meter. As a result, the state of the artexhibits a limited ability to correct for multiphase conditions onmulti-frequency Coriolis meters, particularly in combination withvarying pressure, gas void fraction and reduced frequency. Furthermorethe state of the art multi-frequency Coriolis meters lacks the abilityto output gas void fraction as a measured process parameter.

SUMMARY

In accordance with example embodiments of the present disclosure, amethod for determining parameters for, and application of, models thatcorrect for the effects of fluid inhomogeniety and compressibility onthe ability of Coriolis meters to accurately measure the mass flowand/or density of a process fluid on a continuous basis is disclosed.Example embodiments mitigate the effect of multiphase fluid conditionson a Coriolis meter.

In an example embodiment, at least one measurement of the propagation ofthe speed of sound through a process fluid is employed to determine thegas volume fraction and employed in the interpretation of vibrationalcharacteristics of at least one conduit. Fluid properties, including,process-fluid density, mass flow rate, gas void fraction andliquid-phase density are interpreted using an empirical model in whichparameters of said model are determined using an optimization algorithm.In combination with the measurement of the propagation of the speed ofsound through the process-fluid, at least one conduit is vibrated atmore than one frequency, or more than one conduits are vibrated atdifferent frequencies. Measurement of the mass flow and/or densitythrough at least one conduit vibrated at more than one frequency, ormore than one conduits are vibrated at different frequencies, incombination with the measurement of the speed of sound propagatingthrough the fluid supports an error reducing algorithm that providescontinuous error correction in a flowing fluid of varying properties.

The method and apparatus of the disclosure improves the accuracy ofmulti-frequency Coriolis meters on both homogeneous and non-homogeneousflows. Homogeneous flows include single-phase flows. Moving fluid withentrained gas in which the no significant decoupling occurs due to theinhomogeniety of the bubbles can effectively be treated as homogeneousflows. Measured speed of sound combined with the disclosed method may beused to account for compressibility in both homogeneous flows andnon-homogeneous flows.

The present disclosure describes a method and apparatus that measuresfluid density and mass flow rate with improved accuracy in the presenceof varying levels of compressibility and/or inhomogeneity of a fluidflowing through vibrating conduits, also referred to as vibrating flowtubes.

An example embodiment accurately characterizes multiphase flows within amulti-frequency Coriolis meter. The example embodiment is a method formeasuring process-fluid mass flow and/or density in a Coriolis meterhaving one or more flow tubes, with flow tube vibration occurring at atleast two different resonant frequencies in combination with a measuredsound speed through the process fluid to provide the basis fordetermination of parameters in a model used to correct for the effectsof fluid inhomogeneity and compressibility

An optimization algorithm minimizes an error function to interpret theapparent mass flow rate and apparent densities measured at twofrequencies in terms of the actual mass flow rate and actual density ofa mixture flowing through a Coriolis meter. This error function is basedon equating the interpreted mass flow rates at two frequencies andequating the interpreted process fluid densities. For this example, theerror function is defined as a weighted sum of the square of thenormalized difference in mass flow and densities at the two frequencies,for example:

${error} \equiv {{\alpha_{\overset{.}{m}}\left( {\left( {{\overset{.}{m}}_{f\; 1_{trial}} - {\overset{.}{m}}_{f2_{trial}}} \right)/\left( {{\overset{.}{m}}_{f1_{trial}} + {\overset{.}{m}}_{f2_{trial}}} \right)} \right)}^{2} + {\alpha_{\rho}\left( {\left( {\rho_{f1_{trial}} - \rho_{f2_{trial}}} \right)/\left( {\rho_{f1_{trial}} + \rho_{f2_{trial}}} \right)} \right)}^{2}}$

In the above expression, the trial mass flows and densities are formedby correcting the measured, or apparent, mass flows and densities toactual mass flows and density using an over reading function determinedby a mathematical model which incorporates the effects of process-fluidinhomogeneity and compressibility. The error is minimized by adjustingparameters within the mathematical model such that corrected mass flowsand corrected densities predicted at the two frequencies match,respectively. To determine the parameters of the multiphase flow throughthe meter, the error function is then evaluated over a wide range ofreduced frequency parameters; which in this model influences bothcompressibility and gas void fraction, and the gas damping ratioparameter, also referred to as decoupling parameter. Once the parametersof the model are optimized such that the error function is minimized, f,the optimized mass flow and mixture density are determined by utilizingthe optimized parameters in the model to correct the measuredprocess-fluid mass flow and density. One skilled in the art understandsthat the parameters of a model may be adjusted or optimized. The presentdisclosure may refer to adjusted or optimized interchangeably.

In another example embodiment, an empirical model for the effects ofprocess fluid inhomogeneity and compressibility formulated by Hemp isused, where □_(measured) and {dot over (m)}_(measured) are the densityand mass flow reported by a Coriolis meter operating on a process-fluid,and would be calibrated to accurately represent the density and massflow if the process fluid was a homogeneous fluid with a sufficientlylow, or known, reduced frequency.

ρ_(liquid) and {dot over (m)}_(liquid) are the actual liquid density andmixture mass flow rate. Note that for gas-entrained mixtures, the massflow of the gas phase is typically negligible compared to the mass flowof the liquid, and therefore the mixture mass flow and the liquid massflow rates are essentially identical.

$\frac{\rho_{meas}}{\rho_{liquid}} = {1 - {k_{d}\phi_{g}} + {{G_{d}({f\_ red})}^{2}\mspace{14mu} {for}\mspace{14mu} {density}}}$$\frac{{\overset{.}{m}}_{meas}}{{\overset{.}{m}}_{liquid}} = {1 - {\frac{\left( {k_{m} - 1} \right)}{1 - \phi_{g}}\phi_{g}} + {{G_{m}\left( f_{red} \right)}^{2}\mspace{14mu} {for}\mspace{14mu} {mass}\mspace{14mu} {flow}}}$

Where

$f_{fed} \equiv \frac{\frac{2\pi f_{tube}D}{2}}{a_{mix}}$

the reduced frequency, f_(tube) is the vibrational frequency of flowtube (in Hz) D is the inner flow diameter of said flow tube, and a_(mix)is the mixture speed of sound.

-   -   Gd and Gm are compressibility parameters for the density and        mass flow errors, respectively, and Kd and Km are decoupling        parameters for the density and the mass flow error,        respectively, and □_(g) is the gas void fraction.

Hemp's formulation provides a compact parametric model for correctingfor the effects of inhomogeneity and compressibility on the mass flowand the density as reported by a Coriolis meter, calibrated onhomogeneous process fluids operating at low reduced frequencies, butoperating on process fluids with inhomogeneity and/or significantcompressibility. Hemp's formulation also provides a model thatexplicitly identifies the role of the gas void fraction, □_(g), inquantifying decoupling effects associated with inhomogeneity and thereduced frequency in quantifying compressibility effects. Note thatHemp's model for the influence of inhomogeneity and compressibility isexpressed in terms of gas void fraction and reduced frequency, each ofwhich are readily determined from a process-fluid sound speedmeasurement in conjunction with other information typically availablefrom Coriolis meters and other common process measurements.

The effect of compressibility as a function of frequency is captured inthe model with the reduced frequency. Hemp proposes that thecompressibility constants for density and mass flow are Gd=0.25 andGm=0.5, respectively. The values suggested by Hemp's reduced order modelcan be applied directly, or these values could be determined through anoptimization process. In the first example developed below, we assumethe values for Gd and Gm suggested by Hemp.

As described by Hemp, the decoupling constant for bubbly flows can be afunction of many parameters including bubble size, bubble sizedistribution, gas/liquid density ratio, vibration frequency, and otherparameters, many of which are unknown in many applications. Theliterature indicates that decoupling effects will depend on the inverseStokes number as seen in the following equation:

$k_{d} = {{{F(\delta)}\mspace{14mu} {where}\mspace{14mu} \delta} \equiv \sqrt{\frac{2v_{f}}{\omega a_{bubble}^{2}}}}$

Where F indicaed and undefined function ν_(f) is the kinematicviscosity, v is the vibration frequency of the mode of interest of thevibrating flow tube, and a_(bubble) is the radius of the bubbles.

Based on theory and data presented in literature, it is reasonable toassume that the decoupling constant, Kd, used in the interpretation ofvibrating tube density measurements at two different frequencies, but onthe same fluid, may differ. Additionally, it is reasonable to assumethat the decoupling parameters would vary with varying fluid conditions.

Hemp's formulations rely on decoupling constants, (Kd, Km), and/orcompressibility constants (Gd, Gm), which in general are unknown, anddepend or details of the applications as fluid viscosity, surfacetension, bubble size, tube vibrational frequency, etc. Not only is thisinformation not typically available in most applications, thisinformation likely changes significantly with process fluid variability.

An example embodiment of the disclosure provides a methodology to enablepractical determination of relevant parameters in relevant correctionmodels to mitigate the effects of multiphase conditions on Coriolismeters. Said parameters may be determined during the operation of theCoriolis meter. With the parameters of relevant models identified andupdated as needed, said models can be applied to enable Coriolis metersto practically and accurately measure the mass flow and/or density aswell as other parameters of the process fluid.

The methodology described below provides a method to leverage mass flowand/or density measurements made simultaneously within the same flowtube or at different times but under the same flow conditions, at two ormore frequencies in conjunction with a process fluid sound speed toprovide the basis for near real-time determination of said couplingparameters, and thereby, provides a practical method to accuratelycharacterize multiphase flows within multi-frequency Coriolis meters.

Equating the expressions for the density of the liquid phase of aprocess-fluid as measured by the interpretation of the natural frequencyof two modes of vibration of the process-fluid conduits, yields thefollowing equation:

$\rho_{liq} = {\frac{\rho_{m_{1}}}{1 - {k_{d_{1}}\phi_{g}} + {G_{d}\left( f_{{red}_{1}} \right)}^{2}} = \frac{\rho_{m_{2}}}{1 - {k_{d_{2}}\phi_{g}} + {G_{d}\left( f_{{red}_{2}} \right)}^{2}}}$

Rearranging the expression for the last equality yields the following:

${error}_{{den}_{i}} \equiv \left( {\frac{\rho_{m_{1_{i}}}}{\left( {1 - {k_{d_{1}}\phi_{g_{i}}} + {G_{d}f_{{red}_{1_{i}}}^{2}}} \right)} - \frac{\rho_{m_{2_{i}}}}{\left( {1 - {k_{d_{2}}\phi_{g_{i}}} + {G_{d}f_{{red}_{2_{i}}}^{2}}} \right)}} \right)^{2}$

The above equation may be applied at each instance in time for which theapparent density for each frequency is measured, along with the speed ofsound, gas void fractions and resonant tube frequencies. Errorsdetermined from measurements at multiple instances “i” can be expressedas a summation.

Similarly, following Hemp for the mass flow measurement:

$\begin{matrix}{{\overset{.}{m}}_{liq} = \frac{{\overset{.}{m}}_{m_{1}}}{1 - {\left( {k_{M_{1}} - 1} \right){\phi_{g}/\left( {1 - \phi_{g}} \right)}} + {G_{m}\left( f_{{red}_{1}} \right)}^{2}}} \\{= \frac{{\overset{.}{m}}_{m_{2}}}{1 - {\left( {k_{M_{2}} - 1} \right){\phi_{g}/\left( {1 - \phi_{g}} \right)}} + {G_{m}\left( f_{{red}_{2}} \right)}^{2}}}\end{matrix}\quad$

Rearranging:

${Error_{{\overset{.}{m}}_{\iota}}} \equiv {\left( {{\frac{{\overset{.}{m}}_{m_{1_{i}}}}{1 - {\left( {k_{M_{1}} - 1} \right){\phi_{g_{i}}/\left( {1 - \phi_{g_{i}}} \right)}} + {G_{m}\left( f_{{red}_{1_{i}}} \right)}^{2}}--}\frac{{\overset{.}{m}}_{m_{{2)}i}}}{1 - {\left( {k_{M_{2}} - 1} \right){\phi_{g_{i}}/\left( {1 - \phi_{g_{i}}} \right)}} + {G_{m}\left( f_{{red}_{2_{i}}} \right)}^{2}}} \right)2}$

A composite error function can be defined as:

${Error}_{composite} = {\sum\limits_{i + 1}^{M}\left( {{\alpha_{\overset{.}{m}}{Error}_{{\overset{.}{m}}_{\iota}}} + {\alpha_{\rho}{Error}_{\rho_{i}}}} \right)}$

The example cases developed below utilize the density error function tooptimize the density decoupling parameter and the mass flow errorfunction to optimize the mass flow decoupling parameters, respectively.The composite error function could be used for cases in which arelationship between the mass flow and density decoupling parameterscould be established, for example, if it were assumed that the mass flowand density decoupling parameters were equal, i.e. if kd1=km1, and/orkd2=km2.

It should be noted that the interpretation of a measured process-fluidsound speed in terms of gas void fraction using Wood's equation, or anapproximation thereof, or similar, and requires some knowledge of theprocess-fluid density. Wood's equation for the process fluid speed ofsound of a bubbly liquid can be expressed as:

$\frac{1}{\rho_{mix}a_{mix}^{2}} = {\frac{\phi_{g}}{\rho_{gas}a_{gas}^{2}} + \frac{1 - \phi_{g}}{\rho_{liq}a_{liq}^{2}}}$

The mixture density can be expressed as:

ρ_(mix)=(1−φ_(g))ρ_(liq)+φ_(g)ρ_(gas)

And using an ideal gas law and the expression for the speed of sound ofa polytropic gas: p=ρRT and a_(gas)=√{square root over (KRT)}

And assuming the gas density is much less than the liquid density, andthe compressibility of the gas component is far larger than the liquidcomponent results in the following simplification of Wood's equation:

$\frac{1}{\rho_{mix}a_{mix}^{2}} \cong \frac{\phi_{g}}{\rho_{gas}a_{gas}^{2}}$ρ_(mix) ≅ (1 − ϕ_(g))ρ_(liq)

Which has the solution for gas volume fraction as follows.

$\phi_{g} = \frac{1 - \sqrt{1 - \frac{4{PK}}{\rho_{liq}a_{mix}^{2}}}}{2}$

In the examples developed herein, we minimize the density error functionto determine the density of the liquid phase of a bubbly mixturesmeasured utilizing a two-frequency Coriolis meter at multiple instances.The example simulates “net oil” well test, which the liquid fluiddensity is varying due to varying watercut of the produced liquids andthe gas void fraction is also varying. The liquid density at eachinstance is sought to determine measured watercut.

The minimization of the error function utilizes measured process fluidspeed of sound, process fluid pressure, the measured densities at thetwo frequencies from the Coriolis meter, the measured tube vibrationfrequency, the ratio of specific heats (K) for the gas, and “trial” thedecoupling and compressibility parameters to determine gas void fractionas part of the optimizations process.

The simulation utilizes Hemp's model for density errors to simulate themeasured densities from a two-frequency Coriolis meter operating on amixture 55% to 85% watercut with measured mixture sound speeds of 100 to700 m/sec, operating at a process pressure of 200 psia, with entrainedgas with a ration of specific heats of 1.3, where the density of thewater is 1000 kg/m{circumflex over ( )}3 and the density of the oilphase is 930 kg/m{circumflex over ( )}3. The Coriolis meter had 2 inchdiameter flow tubes which one frequency of vibration at 78 Hz, and onefrequency at 420 Hz, with decoupling constants of Kd1=1.2 and Kd2=2.5and a compressibility parameter of Gd=0.25.

The data for the simulation was simulated at 10 instances in which thewatercut and the speed of sound of the process fluid were selectedrandomly between the listed extremes. The measured values for thedensity measured at the two frequencies, the error factor in themeasured densities due to decoupling and compressibility, the reducedfrequencies and the process-fluid sound speed are plotted versus gasvoid fraction in the graph depicted in FIG. 10.

The graph in FIG. 11 shows the results of the optimization based onequating measured densities at two frequencies showing the densitiesmeasured at two frequencies, the corrected liquid density at eachfrequency, and the actual liquid density versus gas void fraction. Trialvalues for the decoupling parameters, Kd1 and Kd2, were bounded between1 and 3, and the compressibility parameter was fixed at Gd=0.25 for theoptimization process. As shown, the corrected liquid densities matchboth each other and the actual fluid density.

The graph in FIG. 12 shows the same results as the graph in FIG. 11, butwith the densities normalized by the input liquid density versus gasvoid fraction.

The graph in FIG. 13 shows the results of a similar optimization withthe same range of the randomly selected input parameters but with 2%random noise added to the speed of sound measurement after the simulatedmeasured densities were calculated. This figure is presented to show adegree of robustness of the optimization process to noise.

The graph in FIG. 14 shows and example of the Error function based onequating densities measured at two frequencies as a function of trialdecoupling parameters, showing the input values of the decouplingparameters and the Optimized values. As shown, the error functionexhibits a trough for which the difference in the two decouplingparameters is nearly constant. Despite this trough, the minimum of theerror function is located at the input values, provided there is processvariability, i.e measurements from multiple process conditions arerequired for the optimization to define a unique solution for thedensity decoupling parameters. This process variability will likely takethe form of variable gas void fraction or other process variablesobserved at multiple instances.

The mass flow measured at two frequencies can also be used to determinethe mass flow decoupling constants Hemp's models. In the example, thesimulation utilizes Hemp's model for mass flow errors to simulate themeasured mass flows from a two-frequency Coriolis meter operating on amixture 55% to 85% watercut with measured mixture sound speeds of 200 to700 m/sec, operating at a process pressure of 200 psia, with entrainedgas with a ration of specific heats of 1.3, where the density of thewater is 1000 kg/m{circumflex over ( )}3 and the density of the oilphase is 930 kg/m{circumflex over ( )}3. The Coriolis meter had 2 inchdiameter flow tubes which one frequency of vibration at 78 Hz, and onefrequency at 420 Hz, with density decoupling constants of Kd1=1.2 andKd2=2.5 and a density compressibility parameter of Gd=0.25. The massflow decoupling parameters were Km1=2.0 and Km2=2.5 with the mass flowcompressibility parameter of Gm=0.5. The mass flow rate was randomlyvarying between 1.5 and 1.8 kg/sec as described in the graph in FIG. 15.

Note that for the mass flow optimization, the liquid phase densityremains a necessary input for interpreting the measured process fluidspeed of sound, and other parameters of the mixture, in terms of gasvoid fraction. In this simulation, it is assumed that the densitydecoupling and compressibility parameters are known prior to the massflow optimization, by for example, performing a density parameteroptimization prior to the optimization to determine the mass flowdecoupling parameters.

The graph in FIG. 15 shows data simulated at 10 instances in which thewatercut and the speed of sound of the process fluid, and mass flow wereselected randomly between the listed extremes. The plot shows measuredvalues for the mass flow was simulated at the two frequencies, errorfactors for the measured mass flow due to decoupling andcompressibility, the reduced frequencies and the process-fluid soundspeed plotted versus gas void fraction.

The graph in FIG. 16 shows the results of optimization based on equatingmeasured mass flows at two frequencies showing the mass flows measuredat two frequencies, the corrected mass flow at each frequency, and theactual mass flow versus gas void fraction. These results include 2%uncorrelated noise on the speed of sound measurement, applied after thecalculation of the offsets. As shown, the corrected liquid mass flowsfor each frequency match both each other and the actual fluid density

The graph in FIG. 17 shows results of optimization based on equatingmeasured mass flows at two frequencies showing the normalized massflowmeasured at two frequencies, the corrected nondimensional mass flow ateach frequency, and the actual nondimensional mass flow versus gas voidfraction.

The graph in FIG. 18 shows an example of an Error function based onequating mass flows measured at two frequencies showing the input valuesof the decoupling parameters and the optimized values for the decouplingparameters. As shown, the mass flow based error function has a similartrough, indicating a preference for solutions in which the delta betweenthe mass flow decoupling constants for the two frequencies are equal.Again, similar to the density based error function shown in the graphtitled “Example of an Error function based on equating densitiesmeasured at two frequency showing the input values of the decouplingparameters and the Optimized values” the minimal error occurs at thecorrect mass flow decoupling parameters provided that there is somedegree of process variability in the input data.

As developed above, the addition of a speed of sound measurement in aprocess-fluid improves the accuracy of a multiple frequency Coriolismeter operating on either homogeneous flows or nonhomogeneous flows forwhich compressibility effects can be significant and where decouplingeffects are negligible. Decoupling effects approach zero for gasentrained flows for large inverse Stokes numbers, in highly viscousflows or in flows with small bubble sizes. In Hemp's formulation,setting Kd1=Kd2=1 and Km1=Km2=1 eliminates any decoupling between gasand liquid phases.

The compressibility constant in Hemp's formulation can be determined foreach vibrational frequency through an optimization process, similar tothat developed above for the decoupling constants.

Other objects and features will become apparent from the followingdetailed description considered in conjunction with the accompanyingdrawings. It is to be understood, however, that the drawings aredesigned as an illustration and not as a definition of the limits of theinvention.

BRIEF DESCRIPTION OF THE DRAWINGS

To assist those of skill in the art in making and using the disclosedinvention and associated methods, reference is made to the accompanyingfigures, wherein: Example figure descriptions follow:

FIG. 1 is a schematic of an example Coriolis meter employing themeasurement of the speed of sound through a process-fluid;

FIG. 2 is an example embodiment showing a Coriolis meter having aplurality of strain based sensors and an exciter;

FIG. 3a is a graph depicting mass flow and density error reading as afunction of gas damping ratio and reduced frequency;

FIG. 3b is a graph depicting mass flow and density error reading as afunction of gas damping ratio and reduced frequency;

FIG. 4a is a graph depicting dual frequency Coriolis optimizationsimulated with reduced order model;

FIG. 4 ba is a graph depicting dual frequency Coriolis optimizationsimulated with reduced order model;

FIG. 5 is a graph depicting optimization function over decouplingparameter using measured reduced frequency;

FIG. 6 is a detailed cross section depicting decoupling parameters;

FIG. 7 is a cross section depicting decoupling parameters

FIG. 8 is an example Coriolis meter adapted to the embodiment;

FIG. 9 is an example Coriolis meter adapted to the embodiment.

FIG. 10 is a graph depicting reduced frequencies and the process-fluidsound speed plotted versus gas void fraction.

FIG. 11 is a graph that shows the results of the optimization based onequating measured densities at two frequencies.

FIG. 12 is a graph that shows densities normalized by the input liquiddensity versus gas void fraction.

FIG. 13 is a graph that shows the results of a range of the randomlyselected input parameters with 2% random noise added to the speed ofsound measurement.

FIG. 14 is a graph that shows and example of the Error function based onequating densities measured at two frequencies as a function of trialdecoupling parameters.

FIG. 15 is a graph that shows mass flow rate randomly varying between1.5 and 1.8 kg/sec.

FIG. 16 is a graph that shows the results of optimization based onequating measured mass flows at two frequencies.

FIG. 17 is a graph that shows results of optimization based on equatingmeasured mass flows at two frequencies.

FIG. 18 is a graph that shows an example of an Error function based onequating mass flows measured at two frequencies.

DESCRIPTION

Referring to FIG. 1, an example embodiment is depicted in theillustration. An array of strain based sensors 130 are in fluidcommunication with flow tubes of a Coriolis meter. The strain basedsensors are used to calculate the speed of sound propagated through theprocess fluid 134. The flow tubes in the Coriolis meter 132 vibrate attwo or more frequencies. The Coriolis meter electronics interpretvibrational characteristics in terms of apparent mass flow and apparentdensity 136. The calculated speed of sound and measured process-fluidpressure are used to determine a reduced frequency for each vibrationalfrequency and a gas void fraction of the process fluid 137. The□_(apparent) and {dot over (m)}_(apparent) are combined with the speedof sound and reduced frequencies and are then sent to an algorithm thatoptimizes decoupling and/or compressibility parameters to minimize errorbetween mass flows and/or densities measured at different frequencies138. The optimized parameters are used to improve measured mass flowand/or density of the process fluid 140.

Referring to FIG. 2, an example Coriolis meter 100 is shown with anumber of strain based sensors 116 arrayed along one of a pair of flowtubes 110/112. Two pick-off coils (114) are shown to measure the naturalfrequency of the vibrations and twist of the vibrating flow tube. Inother words, the pick-off coils responsive to the vibration and twist ofa vibrating flow tube. An exciter 118 is supported by the frameworksurrounding the Coriolis meter.

Referring to FIG. 3 errors interpreting the mass flow and density errorsfor a Coriolis operating on a multiphase flow with an interpretationmethod applicable to homogeneous fluids operating at low reducedfrequencies is depicted in the paired graphs. The errors were predictedusing a simplified aeroelastic model of a Coriolis meter (ref Gysling)over a range of gas bubble damping parameters (decoupling parameter) andreduced frequency parameter (compressibility parameter). Thesepredictions for the errors due to decoupling and compressibility weremade based on a model that utilizes the speed of sound to calculate gasvoid fractions and reduced frequencies. Since gas void fraction andreduced frequency are strongly linked through the process fluid speed ofsound, this formulation utilizes reduced frequency as the variable thatcaptures both decoupling and compressibility effects within the modelpredictions.

One skilled in the art understands that any empirical or computationalmodel that characterizes the relationship between the measuredvibrational characteristics of the fluid-conveying flow tube, i.e. tubephase shift and tube natural frequency, and the multiphase flowproperties within the meters could be used in similar manner.

In this example the reduced order model of Gysling was used to calculatethe apparent mass flow and density “measured” by a dual frequencyCoriolis meter operating on a bubbly mixture. The first in-vacuumbending frequency of the tube was set to 300 Hz, and the second was setat 1100 Hz. The tube diameter was 2 inches. The simulated operatingconditions for the process fluid for this test case was bubbly mixtureof air and water at ambient pressure with 2% gas void fraction. Theactual mass flow through the meter was set at 4.0 kg/sec and the liquiddensity was set at 1000 kg/m{circumflex over ( )}3. The reducedfrequency of tube 1 is 0.57 and tube 2 is 2.09. The gas damping ratio,termed the decoupling parameter in the model, was set to 0.5 for bothfrequencies. The apparent mass flow and mixture density for tube 1 was4.44 kg/sec and 1038 kg/m{circumflex over ( )}3, and for tube 2, 14.18kg/sec and 1927 kg/m{circumflex over ( )}3.

Referring to the aforementioned equation:

error≡α_({dot over (m)})(({dot over (m)} _(f1) _(trial) −{dot over (m)}_(f2) _(trial) )/({dot over (m)} _(f1) _(trial) +{dot over (m)} _(f2)_(trial) ))²+α_(ρ)((ρ_(f1) _(trial) −ρ_(f2) _(trial) )/(ρ_(f1) _(trial)+ρ_(f2) _(trial) ))²

The trial mass flows and densities are formed by correcting themeasured, or apparent, mass flows and densities to actual mass flows anddensity using the over reading function shown as a surface in Error!Reference source not found. for a given set of trial decoupling andcompressibility parameters. An error is formed based on the trial massflows and trial densities associated with the measured, or apparent,mass flows and densities at the two frequencies. The error is minimizedwhen the corrected mass flows and corrected densities predicted at thetwo frequencies match, respectively for a given set of trial decouplingand compressibility parameters. To determine the parameters of themultiphase flow flowing through the meter, the error function is thenminimized over a suitably wide range of decoupling (zsi_(gas)) andcompressibility parameter (f_(lea)) Once the parameters of the model areadjusted such that the error function is minimized, the optimized massflow and mixture density are determined.

Referring to FIG. 4 the error function, for the example described inFIG. 3 versus the decoupling parameter and reduced frequency of thefirst tube frequency is described in the paired graphs. For thisexample, the weighting of the error contributions for the mass flowerrors and the density errors are set to unity. The left figure showsthe general surface shape, and the right shows the surface viewed fromabove with the color axis limited to highlight the existence of multiplesolutions, for example, combinations of decoupling parameter, zsi_gas,and reduced frequency for which the error function approaches zero Asshown, if the error function is evaluated over a range of reducedfrequencies and coupling parameters, the optimization would beconfounded, and unable to determine either the best reduced frequency orthe best decoupling parameter, and therefore unable to report a uniquemass flow or density.

FIG. 4 also Error! Reference source not found. shows the sameoptimization function using the measured process fluid sound speed andtherefore having a known reduced frequency for each vibrational modeevaluated over a range of the decoupling parameters. As shown, thisadditionally-constrained optimization yields a unique solution for themultiphase parameters in this albeit simplified, yet representative,example, thereby enabling the meter to report a more accurate and robustmeasurement of the mass flow and density of the two phase mixture basedon an optimization process equating the mass flow rates measured at twofrequencies and/or the densities measured at two frequencies.

Referring to FIG. 5 this self-consistency example, the decouplingparameter of 0.5 is identified by the optimization. Using thisidentified decoupling parameter, and the apparent mass flow and densityat either tube frequency, enables meter to report accurate mass flow andmixture density. One skilled in the art understands that Coriolis baseddensity measurements in multiphase flows may be more robust andrepeatable than the Coriolis-based mass flow measurement under the sameconditions. In these cases, it may be beneficial to increase theweighting of the density measurement error contribution in the errorfunction compared to the weighting of the mass flow error contribution.

Referring to FIG. 6 a horizontal cross section of two flow-tubes 110/112is depicted in the illustration. The illustration demonstratesdecoupling inhomogeneous flow. Flow tube 110 has a homogeneous flow 160without entrained gas. The center of mass 161 is in the center of theflow-tube 110. The flow-tube 112 has an inhomogeneous flow 162 withentrained gas that is not homogeneous. The center of mass 163 is not inthe center of the flow-tube.

Referring to FIG. 7, a vertical cross section of a flow-tube is depictedin the illustration. An inhomogeneous gas-entrained flow 164 has avarying density of entrained gas about the flow path. Decoupling of aninhomogeneous fluid is said to occur within a fluid when one phase of afluid vibrates differently than another phase. One skilled in the artunderstands that the effects of decoupling are determined substantiallyas a function of the measured gas void fraction.

Referring to FIG. 8 an example Coriolis meter 200 is shown with a numberof strain based sensors 216 arrayed along one of a pair of flow tubes210/212. One skilled in the art understands that the example may includepick-off coils and an exciter as necessary to generate and measure thenatural frequency of the vibrations and twist of the vibrating flow tubemay also be installed on the example Coriolis meter.

Referring to FIG. 9 an example Coriolis meter 300 is shown with a numberof strain based sensors 316 arrayed along a flow tube 312. One skilledin the art understands that the example may include pick-off coils andan exciter as necessary to generate and measure the natural frequency ofthe vibrations and twist of the vibrating flow tube may also beinstalled on the example Coriolis meter.

While example embodiments have been described herein, it is expresslynoted that these embodiments should not be construed as limiting, butrather that additions and modifications to what is expressly describedherein also are included within the scope of the invention. Moreover, itis to be understood that the features of the various embodimentsdescribed herein are not mutually exclusive and can exist in variouscombinations and permutations, even if such combinations or permutationsare not made express herein, without departing from the spirit and scopeof the invention.

1. A Coriolis flowmeter comprising: at least one conduit for conveying aprocess-fluid having a process parameter to be measured; and excitationcircuitry coupled to said at least one conduit operable to provide atleast two vibration frequencies to said at least one conduit; and asystem for measuring the sound speed of said process-fluid; and a systemfor measuring the vibration characteristics of at least two vibrationalmodes of said at least one conduit; wherein a measurement of a processparameter is interpreted by said sound speed and vibrationcharacteristics of at least two vibrational modes of said at least oneconduit.
 2. The Coriolis flowmeter of claim 1 wherein said excitationcircuitry provides at least two vibration frequencies to one conduit. 3.The Coriolis flowmeter of claim 1 further comprising at least a firstand at least a second conduit for conveying a process-fluid to bemeasured; wherein said excitation circuitry vibrates said at least afirst conduit at a first vibration frequency and said at least a secondconduit at a second vibration frequency.
 4. The Coriolis flowmeter ofclaim 1 wherein said process parameter is density of said process-fluid.5. The Coriolis flowmeter of claim 1 wherein said process parameter ismass flow of said process-fluid.
 6. The Coriolis flowmeter of claim 1wherein said system for measuring the sound speed of said process-fluidis an array of sensors responsive to pressure variations within theprocess fluid—deployed on one or more of said at least one conduit. 7.The Coriolis flowmeter of claim 6 wherein said array of sensorsresponsive to pressure variations are strain-based sensors.
 8. TheCoriolis flowmeter of claim 1 wherein the system for measuring the soundspeed of said process-fluid further comprises: at least one strain basedsensor engaged with at least one conduit; and said at least one strainbased sensor is electronically coupled with a central processor.
 9. TheCoriolis flowmeter of claim 1 wherein the system for measuring thevibration characteristics of said at least one conduit furthercomprises: at least one pick-off coil responsive to the vibration atleast one conduit; and said at least one pick-off coil is electronicallycoupled with a central processor.
 10. The Coriolis flowmeter of claim 1wherein a central processor interprets said sound speed and vibrationalcharacteristics of said at least one conduit vibrating at, at least twovibration frequencies, to provide a measurement of the process-fluiddensity.
 11. The Coriolis flowmeter of claim 1 wherein a centralprocessor interprets said sound speed and vibrational characteristics ofsaid at least one conduit vibrating at, at least two vibrationfrequencies, to provide a measurement of the process-fluid mass flow.12. The Coriolis flowmeter of claim 1 wherein said system that measuresprocess-fluid sound speed is an array of sensors responsive the pressurevariations within the process-fluid deployed on a conduit other thansaid at least one conduit.
 13. The Coriolis flowmeter of claim 1 whereinsaid system that measures process-fluid sound speed determines a measureof gas void fraction of the process-fluid; and said system determines areduced vibration frequency of more than one of the vibrationfrequencies of said at least one conduit.
 14. The Coriolis flowmeter ofclaim 1 wherein said system that measures process-fluid sound speeddetermines a more than one reduced frequency of vibration of said atleast one conduit.
 15. A method for optimizing a process parameter ofthe Coriolis meter of claim 1 comprising: vibrating said at least oneconduit at two or more frequencies; and said two or more frequenciesbeing low or known reduced frequencies; and providing homogeneous flowsthrough said at least one conduit; and measuring said process parameterat said two or more frequencies; and calibrating said Coriolis meter tooperate on the effects of process-fluid variability; and measuring aprocess fluid sound speed; and calibrating said measurement of a processparameter interpreted by said sound speed, wherein an optimized processparameter is determined.
 16. The method of claim 15 wherein: saidprocess parameter is the density of said process-fluid.
 17. The methodof claim 15 wherein: said process parameter is the mass flow of saidhomogeneous flow.
 18. The method of claim 15 wherein: the effects ofprocess-fluid variability is fluid inhomogeneity and/or changes in fluidcompressibility.
 19. The method of claim 15 wherein calibrating saidCoriolis meter to measure the effects of process-fluid variabilityfurther comprises the steps of: correcting measured process parameters;and minimizing the difference between corrected process parameters. 20.A Coriolis meter comprising: a processor; and at least two conduits fortransferring a process-fluid to be measured; and excitation circuitrycoupled to said at least two conduits, and in communication with saidprocessor; and said excitation circuitry operable to vibrate said atleast two conduits at, at least a first vibration frequency and at, atleast a second vibration frequency; and a measurement element capable ofmeasuring the speed of sound waves propagated through saidprocess-fluid; and a measurement element capable of measuring theresultant vibration of said at least one conduit; wherein said processorinterprets the measurement of the speed of sound waves propagatedthrough said process-fluid and the measured resultant vibrationcharacteristics of said at least two conduits operable to vibrate at, atleast a first vibration frequency and at, at least a second vibrationfrequency to provide a measurement of the process parameter.
 21. TheCoriolis meter of claim 20 wherein: said measurement element capable ofmeasuring the speed of sound waves propagated through said process-fluidis an array of strain-based sensors in communication with saidprocessor.
 22. The Coriolis meter of claim 20 wherein: said measurementelement capable of measuring the resultant vibration of said at leastone conduit is at least one pick-off coil.
 23. The Coriolis meter ofclaim 20 wherein the measured speed of sound waves propagated throughsaid process-fluid in said at least one conduit; and the measuredresultant vibration of said at least one conduit, in combination, areinterpreted to provide a measurement of the process-fluid mass flow. 24.The Coriolis meter of claim 20 wherein the measured speed of sound wavespropagated through said process-fluid in said at least one conduit; andthe measured resultant vibration of said at least one conduit, incombination, are interpreted to provide a measurement of theprocess-fluid density.
 25. A Coriolis mass flowmeter comprising: atleast one conduit; and a process-fluid flowing through said conduit; andexcitation circuitry coupled with said conduit; and said excitationcircuitry for vibrating said conduit at a first frequency; and saidexcitation circuitry for vibrating said conduit at a second frequency;and at least one strain sensor for measuring the speed of sound throughsaid process-fluid; wherein the amplitude of said at least one conduit,vibrated by said excitation circuitry, is measured; and the measuredspeed of sound through said process-fluid in said at least one conduit,in combination, are interpreted to provide a measurement of theprocess-fluid mass flow.
 26. A flow metering system comprising: aprocess-fluid in one or more conduits; and actuator(s) for vibratingsaid one or more conduits; and a sensor engaged with said one or moreconduits for determining a process-fluid gas void fraction and reducedfrequency; and a model for interpreting process-fluid mass flow rateand/or density by the vibrational characteristics of said one or moreconduits; and a calibration of said vibrational characteristics that isapplicable for homogeneous flows at a known or sufficiently low reducedfrequency; and a model to correct interpreted process-fluid mass flowrate and/or density based on said calibration, applicable forhomogeneous flows at a known or sufficiently low reduced frequency;wherein the correction terms for the effects of decoupling aredetermined substantially as a function of the measured gas void fractionand the correction terms for the effects of compressibility aredetermined substantially as a function of the measured reducedfrequency.
 27. The system of claim 26, wherein said sensor is an arrayof strain-based sensors.